The Eigenvectors of Single-spiked Complex Wishart Matrices: Finite and Asymptotic Analyses

10/22/2021

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by   Prathapasinghe Dharmawansa, et al.

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Let 𝐖∈ℂ^n× n be a single-spiked Wishart matrix in the class 𝐖∼𝒞𝒲_n(m,𝐈_n+ θ𝐯𝐯^†) with m≥ n, where 𝐈_n is the n× n identity matrix, 𝐯∈ℂ^n× 1 is an arbitrary vector with unit Euclidean norm, θ≥ 0 is a non-random parameter, and (·)^† represents the conjugate-transpose operator. Let 𝐮_1 and 𝐮_n denote the eigenvectors corresponding to the samllest and the largest eigenvalues of 𝐖, respectively. This paper investigates the probability density function (p.d.f.) of the random quantity Z_ℓ^(n)=|𝐯^†𝐮_ℓ|^2∈(0,1) for ℓ=1,n. In particular, we derive a finite dimensional closed-form p.d.f. for Z_1^(n) which is amenable to asymptotic analysis as m,n diverges with m-n fixed. It turns out that, in this asymptotic regime, the scaled random variable nZ_1^(n) converges in distribution to χ^2_2/2(1+θ), where χ_2^2 denotes a chi-squared random variable with two degrees of freedom. This reveals that 𝐮_1 can be used to infer information about the spike. On the other hand, the finite dimensional p.d.f. of Z_n^(n) is expressed as a double integral in which the integrand contains a determinant of a square matrix of dimension (n-2). Although a simple solution to this double integral seems intractable, for special configurations of n=2,3, and 4, we obtain closed-form expressions.